We present an analysis of wave mixing in the recently developed Josephson traveling-wave parametric amplifier (JTWPA). Circuit simulations performed using WRSPICE show the full behavior of the JTWPA, allowing propagation of all tones. The coupled mode equations (CMEs) containing only pump, signal, and idler propagation are shown to be insufficient to completely capture complex mixing behavior in the JTWPA. Extension of the CMEs through additional state vectors in the analytic solutions allows closer agreement with WRSPICE. We consider an ordered framework for the systematic inclusion of extended eigenmodes and make a comparison with WRSPICE at each step. The agreement between the two methods validates both approaches and provides insight into the operation of the JTWPA.